Strong Coresets for k-Median and Subspace Approximation: Goodbye Dimension
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We obtain the first strong coresets for the k-median and subspace approximation problems with sum of distances objective function, on n points in d dimensions, with a number of weighted points that is independent of both n and d; namely, our coresets have size poly(k/ϵ). A strong coreset (1+ϵ)-approximates the cost function for all possible sets of centers simultaneously. We also give efficient nnz(A) + (n+d)poly(k/ϵ) + (poly(k/ϵ)) time algorithms for computing these coresets. We obtain the result by introducing a new dimensionality reduction technique for coresets that significantly generalizes an earlier result of Feldman, Sohler and Schmidt FSS13 for squared Euclidean distances to sums of p-th powers of Euclidean distances for constant p>1.
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